Problem: The value of Vishal's car is depreciating exponentially. The relationship between $V$, the value of his car, in dollars, and $t$, the elapsed time, in years, since he purchased the car is modeled by the following equation. V = 22,500 ⋅ 10 − t 12 V=22{,}500 \cdot 10\^{{-\frac{t}{12}}} How many years after purchase will Vishal's car be worth $\$10{,}000$ ? Give an exact answer expressed as a base- $10$ logarithm. years
Explanation: Thinking about the problem We want to know how many years, $t$, it will take for the value of Vishal's car, $V$, to be $\$10{,}000$. So we need to find the value of $t$ for which $V=10{,}000$. Substituting $10{,}000$ in for $V$ in the model gives us the following equation: 10,000 = 22,500 ⋅ 10 − t 12 10{,}000=22{,}500 \cdot10\^{{-\frac{t}{12}}} Solving the equation We can solve the equation as shown below. 22,500 ⋅ 10 − t 12 10 − t 12 − t 12 t = 10,000 = 4 9 = log ( 4 9 ) = − 12 log ( 4 9 ) \begin{aligned}22{,}500\cdot 10\^{{-\frac{t}{12}}}&=10{,}000\\\\ 10\^{{-\frac{t}{12}}}&=\dfrac{4}{9}\\\\ -\dfrac{t}{12}&=\log\left(\dfrac{4}{9}\right)\\\\ t&=-12\log\left(\dfrac{4}{9}\right)\\\\ \end{aligned} It will take $-12\log\left(\dfrac{4}{9}\right)$ years for the value of Vishal's car to depreciate to $\$10{,}000$. The expression above represents an exact solution to the equation. We can use a calculator to approximate the value of the expression, but this will be a rounded inexact answer. The answer The answer is $-12\log\left(\dfrac{4}{9}\right)$ years.